302 XIV INTEGRATION IN LOCALLY COMPACT GROUPS sequence (mj(Y(aj)fj — //)). (It is enough to prove that if the sequence of functions (%(Y(^)/} ~~ /})) converges to a function F, then D*/ exists and is equal to F. Fix a neighborhood W of e such that ||y(z)F — F|| < Jfi for all z e W, then fix a real number 8 > 0 such that a$ e W for 0 ^ k < 8 w^ . Now show that for large y, and k<Smj, we have and deduce that ||r-1(YWO)/'-/) - F|| ^ e for 0 g r < S.) In particular, if the sequence (<2jrmj]) tends to e for all r e R, then the sequence OW/(Y(«J)./} — /})) tends uniformly to 0. 14. (a) Consider the sequence (gt) constructed in Problem 1 l(c). Show that the sequence of numbers ///*/ is bounded above. (Argue by contradiction, by supposing that there exists a subsequence (/(#))* «N sucn tnat ^m ^w/'W = 0. Reduce to the case where ft-* 00 the sequence (g^)) converges uniformly to a limit g in G, and there exists a sequence of points cf(fc) e U"(^) converging to an element c ^ e. Then there exists an element flic*) e Uf<*) such that Now use the hypothesis lim ntwlf(k) = 0, Problem 7 of Section 12.9, and the definition *-KX> of the Uj to deduce that for each r e R the sequence («^JJ<fc)3) tends to e. Using Problem 13(b), show that this implies that \\v(c)g—g\\ = 0, contrary to the fact that g is adequate.) (b) Show that if g is the uniform limit of a subsequence of the sequence (g{\ then g is X-differentiable for every one-parameter subgroup X. (Using (a) and Problem ll(c), show that the functions i(t(X(\li))gi — gi) form a uniformly bounded and uniformly equicontinuous set, and apply Problem 13(b).) 15. Let G be a unimodular locally compact group and ft a Haar measure on G. Show that a function /e -^cK^ j^) ^s equal almost everywhere to a function which is uniformly continuous with respect to a right-invariant distance on G if and only if the function s i— *• y(s)f, with values in «^c(^> /?)» is continuous at the point e. (To show that the condition is sufficient, consider a sequence (un) of functions belonging to «#" +(G) such that Supp(wn)c vn (notation of (14.11.2)) and | un(x) df$(x) = 1. Prove that Noo(wn */—•/) tends to 0 with I///, by using (13.17.1) and the Lebesgue-Fubini theorem.) 16. If p. is any bounded measure on R, the integral is defined for every z e C such that J>'z ^ 0, and is an analytic function of z in each of the half-planes ,/z > 0, Jz < 0. It is analytic also at all x 6 R which do not lie in the support of fji. The function F^ is called the Stieltjes transform of ju. (a) Let x0 e R. Suppose that the restriction of /x to an open neighborhood V of 0 has a continuous density g with respect to Lebesgue measure on V. Show that as y tends to 0 through positive values, >(FM(x0 4- /») tends to a limit equal to 7rg(x0).d. Then, for each neighborhood U of e, there exists a number e > 0 such that